: Theorem: All polynomial sequences of binomial type are of this form.

: Theorem ( A. Korselt 1899 ): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of, it is true that ( where means that divides ).

For larger values of n, Fermat's Last Theorem states there are no positive integer solutions ( x, y, z ).</ td >

Theorem 1: If a property is positive, then it is consistent, i. e., possibly exemplified.

With the publication of Gold's Theorem 1967 it was claimed that grammars for natural languages governed by deterministic rules could not be learned based on positive instances alone.

Theorem For any normalized continuous positive definite function f on G ( normalization here means f is 1 at the unit of G ), there exists a unique probability measure on such that

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea del Centina, “ she had actually shown that it holds for every exponent p < 197 .” L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

Using Wilson's Theorem, for any odd prime we can rearrange the left hand side of

* December 25-Fermat claims a proof of the theorem on sums of two squares in a letter to Mersenne (" Fermat's Christmas Theorem "): an odd prime p is expressible as the sum of two squares.

The odd Laplacian measures the failure of Liouville's Theorem.

In that case, there exists an odd Darboux Theorem.

Theorem: A group decision function with an odd number of voters meets conditions 1, 2, 3, and 4 if and only if it is the simple majority method.

For example, there are 20, 138, 200 Carmichael numbers between 1 and 10 < sup > 21 </ sup > ( approximately one in 50 billion numbers ).< ref name =" Pinch2007 "> Richard Pinch, " The Carmichael numbers up to 10 < sup > 21 </ sup >", May 2007 .</ ref > This makes tests based on Fermat's Little Theorem slightly risky compared to others such as the Solovay-Strassen primality test.

On the other hand, the scholars invoking Gödel's Theorem appear, at least in some cases, to be referring not to the underlying rules, but to the understandability of the behavior of all physical systems, as when Hawking mentions arranging blocks into rectangles, turning the computation of prime numbers into a physical question.

This property enables-adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles.

*" Theorem 1 is the famous theorem of Cantor, that the real numbers are uncountable.

The Lehmers also assisted Harry Vandiver with his work on Fermat's Last Theorem, computing many Bernoulli numbers required.

The 2-dimensional Gauss – Bonnet Theorem arises as the special case where the analytical index is defined in terms of Betti numbers and the topological index is defined in terms of the Gauss – Bonnet integrand.

It is widely believed that Kummer was led to his " ideal complex numbers " by his interest in Fermat's Last Theorem ; there is even a story often told that Kummer, like Lamé, believed he had proven Fermat's Last Theorem until Dirichlet told him his argument relied on unique factorization ; but the story was first told by Kurt Hensel in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources.

* Ideal Numbers, Proof that the theory of ideal numbers saves unique factorization for cyclotomic integers at Fermat's Last Theorem Blog.

Hutchings's Theorem states that any of the prime numbers 5, 7, 11, 13, …, wins as a first move, but very little is known about the subsequent winning moves: these are the only winning openings known.

In it she showed that the theory of the rational numbers was undecidable by showing that elementary number theory could be defined in terms of the rationals, and elementary number theory was already known to be undecidable ( this is Gödel's first Incompleteness Theorem ).

Then Hahn's Embedding Theorem reduces to Hölder's theorem ( which states that a linearly ordered abelian group is Archimedean if and only if it is a subgroup of the ordered additive group of the real numbers ).

:: If, then by Fermat's Little Theorem each of the numbers is congruent to one modulo.

* The Murderous Maths of Everything, ISBN 1-407-10367-9 ( prime numbers, Sieve of Eratosthenes, Pythagoras ' Theorem, triangle numbers, square numbers, the International Date Line, geometry, geometric constructions, topology, Mobius strips, curves ( conic sections and cycloids Golomb Rulers, 4 dimensional " Tic Tac Toe ", The Golden Ratio, Fibonacci sequence, Logarithmic spirals, musical ratios, Theorems ( including Ham sandwich theorem and Fixed point theorem ), probability ( cards, dice etc.

* Imaginary numbers, A review of The Parrot's Theorem by Simon Singh

* Imaginary numbers, A review of The Parrot's Theorem by Simon Singh.

The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.

Since Af are distinct prime polynomials, the polynomials Af are relatively prime ( Theorem 8, Chapter 4 ).

If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.

Theorem: If K < sub > 1 </ sub > and K < sub > 2 </ sub > are the complexity functions relative to description languages L < sub > 1 </ sub > and L < sub > 2 </ sub >, then there is a constant c – which depends only on the languages L < sub > 1 </ sub > and L < sub > 2 </ sub > chosen – such that

* James ` s Theorem For a Banach space the following two properties are equivalent:

: Theorem XXX: " The following classes of partial functions are coextensive, i. e. have the same members: ( a ) the partial recursive functions, ( b ) the computable functions.

Therefore both and are inverses of By Theorem 1. 5,

They are expressed using the Reynolds Transport Theorem.

In a ring all of whose ideals are principal ( a principal ideal domain or PID ), this ideal will be identical with the set of multiples of some ring element d ; then this d is a greatest common divisor of a and b. But the ideal ( a, b ) can be useful even when there is no greatest common divisor of a and b. ( Indeed, Ernst Kummer used this ideal as a replacement for a gcd in his treatment of Fermat's Last Theorem, although he envisioned it as the set of multiples of some hypothetical, or ideal, ring element d, whence the ring-theoretic term.

Assume p and q − 1 are relatively prime, a similar application of Fermat's Little Theorem says that ( q − 1 )< sup >( p − 1 )</ sup > ≡ 1 ( mod p ).

Pavel Samuilovich Urysohn, Pavel Uryson () ( February 3, 1898, Odessa – August 17, 1924, Batz-sur-Mer ) was a Jewish mathematician who is best known for his contributions in the theory of dimension, and for developing Urysohn's Metrization Theorem and Urysohn's Lemma, both of which are fundamental results in topology.

: Theorem: Armstrong's axioms are sound and complete ; given a header and a set of FDs that only contain subsets of, if and only if holds in all relation universes over in which all FDs in hold.

The Second Main Theorem, more difficult than the first one tells that there are relatively few values which the function assumes less often than average.

This classical statement, as well as the classical Divergence theorem and Green's Theorem, are simply special cases of the general formulation stated above.

An excellent example is Fermat's Last Theorem, and there are many other examples of simple yet deep theorems in number theory and combinatorics, among other areas.

The Coase Theorem states that assigning property rights will lead to an optimal solution, regardless of who receives them, if transaction costs are trivial and the number of parties negotiating is limited.

For Theorem I the data are generated in full range, while in Theorem II data is only generated when it surpasses a certain threshold, called Peak Over Threshold models ( POT ).

The Modigliani-Miller Theorem describes conditions under which corporate financing decisions are irrelevant for value, and acts as a benchmark for evaluating the effects of factors outside the model that do affect value.

The Myhill-Nerode Theorem for tree automaton states that the following three statements are equivalent:

Other Merton alumni are Bodleian Library founder Thomas Bodley, the Oxford Calculators, Director-General of the BBC Mark Thompson and Sir Andrew Wiles who proved Fermat's Last Theorem.

: Task Completion Theorem: Nevertheless, some tasks are completed, since the intervening presence is itself attempting a task and is, of course, subject to interference.